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June  2020, 25(6): 2203-2221. doi: 10.3934/dcdsb.2019224

Density dependent replicator-mutator models in directed evolution

IMAG, Université de Montpellier, CNRS, Montpellier, 34000, France

* Corresponding author: Matthieu Alfaro

Received  January 2019 Revised  May 2019 Published  September 2019

We analyze a replicator-mutator model arising in the context of directed evolution [24], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [14] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.

Citation: Matthieu Alfaro, Mario Veruete. Density dependent replicator-mutator models in directed evolution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2203-2221. doi: 10.3934/dcdsb.2019224
References:
[1]

M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934.  doi: 10.1137/140979411.  Google Scholar

[2]

M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327.  doi: 10.1090/proc/13669.  Google Scholar

[3]

M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14 (2017), 343-358.  doi: 10.4310/DPDE.2017.v14.n4.a2.  Google Scholar

[4]

M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator–Mutator Equations, J. Dynam. Differential Equations, 2018. Google Scholar

[5]

F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998. Google Scholar

[6]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[7]

V. N. Biktashev, A simple mathematical model of gradual darwinian evolution: emergence of a gaussian trait distribution in adaptation along a fitness gradient, J. Math. Biol., 68 (2014), 1225-1248.  doi: 10.1007/s00285-013-0669-3.  Google Scholar

[8]

I. Bomze and R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11 (1995), 146-172.  doi: 10.1006/game.1995.1047.  Google Scholar

[9]

R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351.  doi: 10.1007/BF00275642.  Google Scholar

[10]

R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91 (1988), 67-83.  doi: 10.1016/0025-5564(88)90024-7.  Google Scholar

[11]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  doi: 10.1007/BF01215194.  Google Scholar

[12]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[13]

W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168.  doi: 10.1137/0136014.  Google Scholar

[14]

M.-E. GilF. HamelG. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561.  doi: 10.1137/16M1108224.  Google Scholar

[15]

M.-E. Gil, F. Hamel, G. Martin and L. Roques, Dynamics of Fitness Distributions in the Presence of a Phenotypic Optimum: An Integro-differential Approach, HAL preprint, 2018. Google Scholar

[16]

K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41 (1981), 1-7.  doi: 10.1137/0141001.  Google Scholar

[17]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, London Mathematical Society Student Texts, Cambridge University Press, 1988.  Google Scholar

[18]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731–736. Google Scholar

[19]

M. NowakN. Komarova and P. Niyogi, Evolution of universal grammar, Science, 291 (2001), 114-118.  doi: 10.1126/science.291.5501.114.  Google Scholar

[20]

K. Page and M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219 (2002), 93-98.  doi: 10.1016/S0022-5193(02)93112-7.  Google Scholar

[21]

P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[22]

P. Stadler and P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30 (1992), 597-631.  doi: 10.1007/BF00948894.  Google Scholar

[23]

P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[24]

A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017. Google Scholar

show all references

References:
[1]

M. Alfaro and R. Carles, Explicit solutions for replicator-mutator equations: Extinction versus acceleration, SIAM J. Appl. Math., 74 (2014), 1919-1934.  doi: 10.1137/140979411.  Google Scholar

[2]

M. Alfaro and R. Carles, Replicator-mutator equations with quadratic fitness, Proc. Amer. Math. Soc., 145 (2017), 5315-5327.  doi: 10.1090/proc/13669.  Google Scholar

[3]

M. Alfaro and R. Carles, Superexponential growth or decay in the heat equation with a logarithmic nonlinearity, Dyn. Partial Differ. Equ., 14 (2017), 343-358.  doi: 10.4310/DPDE.2017.v14.n4.a2.  Google Scholar

[4]

M. Alfaro and M. Veruete, Evolutionary Branching Via Replicator–Mutator Equations, J. Dynam. Differential Equations, 2018. Google Scholar

[5]

F. H. Arnold, Design by Directed Evolution, Acc. Chem. Res, 1998. Google Scholar

[6]

I. Białynicki-Birula and J. Mycielski, Nonlinear wave mechanics, Ann. Physics, 100 (1976), 62-93.  doi: 10.1016/0003-4916(76)90057-9.  Google Scholar

[7]

V. N. Biktashev, A simple mathematical model of gradual darwinian evolution: emergence of a gaussian trait distribution in adaptation along a fitness gradient, J. Math. Biol., 68 (2014), 1225-1248.  doi: 10.1007/s00285-013-0669-3.  Google Scholar

[8]

I. Bomze and R. Burger, Stability by mutation in evolutionary games, Games Econom. Behav., 11 (1995), 146-172.  doi: 10.1006/game.1995.1047.  Google Scholar

[9]

R. Bürger, On the maintenance of genetic variation: Global analysis of Kimura's continuum-of-alleles model, J. Math. Biol., 24 (1986), 341-351.  doi: 10.1007/BF00275642.  Google Scholar

[10]

R. Bürger, Mutation-selection balance and continuum-of-alleles models, Math. Biosci., 91 (1988), 67-83.  doi: 10.1016/0025-5564(88)90024-7.  Google Scholar

[11]

R. Bürger, Perturbations of positive semigroups and applications to population genetics, Math. Z., 197 (1988), 259-272.  doi: 10.1007/BF01215194.  Google Scholar

[12]

R. Carles and I. Gallagher, Universal dynamics for the defocusing logarithmic schrödinger equation, Duke Math. J., 167 (2018), 1761-1801.  doi: 10.1215/00127094-2018-0006.  Google Scholar

[13]

W. H. Fleming, Equilibrium distributions of continuous polygenic traits, SIAM J. Appl. Math., 36 (1979), 148-168.  doi: 10.1137/0136014.  Google Scholar

[14]

M.-E. GilF. HamelG. Martin and L. Roques, Mathematical properties of a class of integro-differential models from population genetics, SIAM J. Appl. Math., 77 (2017), 1536-1561.  doi: 10.1137/16M1108224.  Google Scholar

[15]

M.-E. Gil, F. Hamel, G. Martin and L. Roques, Dynamics of Fitness Distributions in the Presence of a Phenotypic Optimum: An Integro-differential Approach, HAL preprint, 2018. Google Scholar

[16]

K. Hadeler, Stable polymorphisms in a selection model with mutation, SIAM J. Appl. Math., 41 (1981), 1-7.  doi: 10.1137/0141001.  Google Scholar

[17]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems: Mathematical Aspects of Selection, London Mathematical Society Student Texts, Cambridge University Press, 1988.  Google Scholar

[18]

M. Kimura, A stochastic model concerning the maintenance of genetic variability in quantitative characters, Proc. Natl. Acad. Sci., USA, 54 (1965), 731–736. Google Scholar

[19]

M. NowakN. Komarova and P. Niyogi, Evolution of universal grammar, Science, 291 (2001), 114-118.  doi: 10.1126/science.291.5501.114.  Google Scholar

[20]

K. Page and M. Nowak, Unifying evolutionary dynamics, J. Theoret. Biol., 219 (2002), 93-98.  doi: 10.1016/S0022-5193(02)93112-7.  Google Scholar

[21]

P. Schuster and K. Sigmund, Replicator dynamics, J. Theoret. Biol., 100 (1983), 533-538.  doi: 10.1016/0022-5193(83)90445-9.  Google Scholar

[22]

P. Stadler and P. Schuster, Mutation in autocatalytic reaction networks, J. Math. Biol., 30 (1992), 597-631.  doi: 10.1007/BF00948894.  Google Scholar

[23]

P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci., 40 (1978), 145-156.  doi: 10.1016/0025-5564(78)90077-9.  Google Scholar

[24]

A. Zadorin and Y. Rondelez, Natural selection in compartmentalized environment with reshuffling, arXiv preprint, arXiv: 1707.07461, 2017. Google Scholar

Figure 1.  Evolution of Gaussian solutions for $ {\sigma ^2} = 1 $, $ a_0 = 1 $ and (from left to right) $ m_0 = -4 $, $ m_0 = 0 $ and $ m_0 = 4 $
Figure 2.  (A): The first four approximations, for $ 0\leq t\leq 3 $, of the nonlocal term $ \overline{u}(t) $ computed via the fixed point iteration (28). (B): Numerical solution obtained by the method described in Section 5, starting from $ u_0 = \mathbb{1}_{[1/2,3/2]} $, with $ {\sigma ^2} = 1 $. The red points are the points on the graph $ u(t,\cdot) $ with abscissa $ x = \overline u(t) $. The green points are the maxima of $ u(t,\cdot) $. This reveals the dissymmetry of the solution
Figure 3.  Vector field defined by the differential system (31) with $ {\sigma ^2} = 1 $, describing the dynamics of Gaussian solutions. In yellow, the set of initial conditions for which $ a $ blows up in finite time $ T^{\star} $ and in red, dark blue and light blue those for which both $ a $ and $ m $ are globally defined. The red dashed curve is the set of values defined by $ m_0 = -1/{(a_0\sqrt{2{\sigma ^2}})} $, for which $ a $ tends to infinity and $ m $ tends to zero as time goes to infinity. The dark blue region corresponds to the values leading to an anti-diffusion/diffusion behaviour. The light blue region corresponds to the values leading to a pure diffusion behaviour
Figure 4.  Case $ (i) $ concentration in finite time. The values of the parameters are $ a_0 = 5/64 $, $ m_0 = -585/64 $, $ {\sigma ^2} = 1 $. It follows that $ T^\star\approx 1.878 $ and $ m(T^\star)\approx -1.277<0 $
Figure 5.  Case $ (ii) $ concentration in infinite time: the solution converges to a Dirac mass at zero. The values of the parameters are $ a_0 = 3/16 $, $ m_0 = -8\sqrt{2}/3 $, $ {\sigma ^2} = 1 $
Figure 6.  Case $ (iii) $ with $ m_0<0 $, anti-diffusion/diffusion behaviour. The values of the parameters are $ a_0 = 2/10 $, $ m_0 = -34/10 $, $ {\sigma ^2} = 1 $
Figure 7.  Case $ (iii) $ with $ m_0\geq 0 $, the solution is flattening and accelerating. The values of the parameters are $ a_0 = 3/2 $, $ m_0 = 7/2 $, $ {\sigma ^2} = 1 $
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